Six sixes

Digits and numbers are wonderful toys. The number of problems with them surpasses the imagination. Still, great minds think alike.
In 2013 I posed a classical problem to make as many numbers as possible by combining four fours.

In March I posed the challenge to make all the numbers of 0 to 20 by combining 5 5’s.

1) 6 6’s^***/*****
Today’s problem is a sequel: make all the numbers 0 till 20 using 6 6’s. You can use brackets, +,-,*,/, ^and sqrt. You msy use ! for just one of the numbers.
You can check your solutions here and here

When googling, I found the puzzle masters at occupymath have posed the same challenge.

2) 7 7’s^***/*****
Of course the problem can be generalized to 7 7’s. Make all numbers 0-24 using seven sevens, brackets, +-*/, no faculty.
You can check your solutions here and here

3) general ^***/*****
The next problem is not, as you may have thought, to make all numbers 0-20 with 8 8’s. Rather the question is, can the numbers 0-20 with made with all digits? In base 11, in base 12?
For a start:
0 = (n-n)*(n+…+n)
1 = n/n + (n-n)*(n+…+n)
2 = n/n + n/n + (n-n)*(n+ … + n)
and so on.

What’s next?

1) puzzle 1^****/*****
What is next in the sequence 1, 2, 3, 8, 15, 48, 105, 384?

2) puzzle 2^****/*****
What is next after 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880?

If you are baffled, you can find the solutions here.

Bongard dates (3)

In 1967 the Russian scientist M.M. Bongard published a book containing 100 problems. Each problem consists of 12 small boxes: six boxes on the left and six on the right. Each of the six boxes on the left conforms to a certain rule. Each box on the right contradicts this rule. Your task, of course, is to figure out the rule.

1)Bongard problem dates 5^***/*****

2)Bongard problem dates 6^****/*****

The original Bongard problems were geometrical and thus, in theory, culture free. These dates are western dates, and thus not culture independent. I have used the Italian/Dutch format. The 2-weekly puzzle column in the Guardian in the past already expanded the scope from geometry to language, but as far as I know the dates are a new territory. During my recent visit to Burkina Faso I wrote up 12 problems, so I have enough to trouble you the first half of 2020.

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.

Happy puzzling!